I would like to know what is the most general form of the Gauss-Bonnet theorem in the plane for curves. It is well known for that for any piecewise $C^2$ simply connected curve with corners, one has

$\int_{\partial \Omega} \kappa(y) dS(y) = 2\pi$,

where $\kappa$ denotes the curvature of the boundary $\partial \Omega$. This formula continues to hold for any curve for which $\hat n \cdot (1,0)$ (for instance) defines a $BV$ function.

Question: I would like to know to what extent this generalizes. For example, if $\Omega$ is a simply connected, compact set in $\mathbb{R}^2$ whose boundary has finite length and has generalized curvature $\kappa \in L^{\infty}(\partial \Omega)$, does it hold that $\int_{\partial \Omega} \kappa(y) dS(y) = 2\pi$? If not, are there counter examples in the plane where this fails?